Saute' onions and mushrooms in 1 TBSP of butter until tender.Remove from heat.
Stir in chicken, cheese eggs, parsley, salt, and tarragon until well-blended.
Spread 1 sheet of phyllo on smooth work surface and brush lightly with some of remaining butter.
Layer a second sheet of phyllo over the first and repeat with lightly brushing butter over the phyllo sheet.
Spoon about 2/3 cup of chicken mixture along the narrow end of the phyllo.
Fold in the sides of the phyllo and roll loosely.
Place seam-side down on greased baking sheet and brush top with butter.
Repeat to make 8 chicken rolls in all.
Enjoy!